Detecting and creating oscillations using multifractal methods

By comparing the Hausdorff multifractal spectrum with the large deviations spectrum of a given continuous function f , we find sufficient conditions ensuring that f possesses oscillating singularities. Using a similar approach, we study the nonlinear wavelet threshold operator which associates with any function f = ∑ j ∑ kd j,k ψ j,k ∈ L 2 (ℝ) the function series f t whose wavelet coefficients are d tj,k = d j,k 1   | d  j,k|≥2   – jγ, for some fixed real number γ > 0. This operator creates a context propitious to have oscillating singularities. As a consequence, we prove that the series f t may have a multifractal spectrum with a support larger than the one of f . We exhibit an example of function f ∈ L 2 (ℝ) such that the associated thresholded function series f t effectively possesses oscillating singularities which were not present in the initial function f . This series f t is a typical example of function with homogeneous non‐concave multifractal spectrum and which does not satisfy the classical multifractal formalisms. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)